Prove the following demorgan's laws: if latex: xx, latex: aa and latex: bb are sets and latex: \{a_i: i\in i\} {ai: i∈i} is a family of sets, then
latex: x-(a\cup b)=(x-a)\cap (x-b)
latex: x-(\cup_{i\in i}a_i)=\cap_{i\in i}(x-a_i)

I'll assume the usual definition of set difference, .

Let . Then and . If , then and . This means and , so it follows that . Hence .

Now let . Then and . By definition of set difference, and . Since , we have , and so . Hence .

The two sets are subsets of one another, so they must be equal.

The proof of this is the same as above, you just have to indicate that membership, of lack thereof, holds for all indices .

Proof of one direction for example:

Let . Then and , which in turn means for all . This means , and , and so on, where , for all . This means , and , and so on, so . Hence .